AC-09-C1.5.9 ABSTRACT

Transcription

1 AC-9-C.5.9 AENT-BASED MODELIN TECHNIQUES FOR CONTROL OF SATELLITE IMAIN ARRAYS IN MULTI-BODY REIMES Lindsay D. Millard Adjunct Researcher School of Aeronautics and Astronautics Purdue University West Lafayette, Indiana Kathleen C. Howell Hsu Lo Professor of Aeronautical and Astronautical Engineering School of Aeronautics and Astronautics Purdue University West Lafayette, Indiana ABSTRACT Decentralized control and agent-based modeling techniques are employed to determine optimal spacecraft motion for formations having multifaceted objectives in complex dynamic regimes. Cooperative satellite agents share a common objective (high resolution imaging and simultaneously pursue private goals (minimal fuel usage. An algorithm is developed based on dual decomposition. The solution from an agent-based model is compared to a traditional, non-linear optimal control solution. Then, by exploiting the reduced computational requirements, the agentbased model simulates arrays with increasing numbers of satellites and constraints. INTRODUCTION Imaging a single planet traveling around a star, hundreds of light years from Earth, is a daunting task. Traditional Earth-based telescopes must continuously expand in size to resolve increasingly distant and faint objects. Unfortunately, as the size of a telescope increases, it becomes too heavy and too expensive to launch, thus motivating alternative designs for deepspace observers. One such design is a free-flying array or constellation of smaller satellites that work together to form a single optical platform. Recent NASA mission concepts include satellite constellations to facilitate imaging and identification of distant planets. These concepts are diverse, encompassing designs such as the Terrestrial Planet Finder-Occulter (TPF-O, where a monolithic telescope is aided by a single occulter spacecraft, and the Micro-Arcsecond X-Ray Interferometry Mission (MAXIM, where as many as sixteen spacecraft move together to form a space interferometer,,. In most design concepts, however, the location of the formation is specified to be near one of the Sun-Earth collinear libration points. Thus, when modeling the motion of the deep-space imaging formations, it is advantageous to employ a multi-body gravitational model. 4 The complexity of a nonlinear multi-body model for spacecraft motion coupled with sophisticated imaging objectives has limited the use of traditional optimal control theory in determining spacecraft motion within imaging arrays. The problem of ascertaining the optimal motion of spacecraft for interferometric imaging (while minimizing fuel has been solved for specific and simplified cases. Hussein, Scheeres, and Hyland 5,6,7 identify a special class of two-dimensional spiral motions that satisfy the necessary imaging requirements for certain interferometric imaging missions. Isolation of this class of motion serves as a basis for the identification of a relationship between image quality and fuel expenditure. The spacecraft are assumed to be moving in free space; the only forces acting on the system are control forces. Hussein et al. 8,9 expand this analysis by formulating an optimal control problem to minimize measurement noise and fuel while maximizing certain image metrics. Although the two-point boundary value problem remains unsolved, necessary conditions for optimality are defined and these reveal the general behavior of an optimal solution: a spiral-like motion. Again, the spacecraft are assumed to be moving in free space. Finally, Hussein and Bloch, apply dynamic coverage optimal control (a geometric control method to interferometric imaging arrays. However, the formulation incorporates assumptions that include: a system that evolves only on planar or paraboloidal Riemannian manifolds and is not evolving in any gravitational field. Most recently, Millard and Howell formulate a nonlinear optimal control problem to maximize resolution while minimizing fuel for interferometric spacecraft imaging formations. The problem is solved numerically for spacecraft moving under the influence of two large gravitational bodies, as modeled in the circular-restricted threebody problem. Due to limitations in computational capability, the analysis does not include formations of

2 satellites containing more than two spacecraft. However, the analytical formulation of the problem does apply to infinitely larger formations. This study builds on the results of Millard and Howell. The high complexity inherent in the analysis of spacecraft imaging formations in multi-body regimes motivates a change in either the analytical formulation of the problem or a transition to more robust computational algorithms and utilities, such as FORTRAN, UNIX, and parallel processing. This paper explores the former possibility by employing decentralized control and agent-based modeling techniques to solve for the optimal motion of spacecraft formations with highly complex objectives, defined in multi-body regimes. Cooperative satellite agents share a common objective (high resolution imaging and simultaneously pursue private goals (minimal fuel usage. An algorithm is developed based on dual decomposition techniques.,4,5,6 The dual problem of an artificially decomposed version of the primal problem is solved, replacing one large computationally intractable problem with many smaller tractable problems. Thus, this method may be most useful when traditional analytical and computational tools are reduced in efficacy due to problem complexity. Simple rules are defined that govern the motion of individual satellites, while complex emergent behavior is observed in formation motion, as a whole. The outcome of the agent-based model is compared to a more traditional non-linear optimal control solution for a relatively simple twosatellite array example. Then, taking advantage of the reduced computational requirement, the agent-based model is employed to simulate more complex arrays with an increasing number of satellites and constraints. It is important to note that agent-based control of satellite arrays has been explored extensively from the perspective of software development, control system hierarchy, communication structure, autonomous control system stability, and cost 7. The TeamAgent software system, based on ObjectAgent technology, has been designed to enable multiple satellites to cooperate autonomously with particular focus on formation flying 8. eneralized agent-based software architecture for formation flying has been devised and is being applied to the Air Force Research Laboratory s TechSat technology demonstration mission. In contrast, this study does not focus on autonomous agent-based control but rather on the solution to optimization problems via dual decomposition and agent-based modeling. SYNTHETIC APERTURE IMAIN MODEL If fuel usage is ignored, motion can be defined for a satellite telescope formation that maximizes its capability to detect or image a distant object. The nature of the optimal motion will vary depending on the design of the satellite imaging system. For example, the optimal planet-finding configuration for TPF-O occurs when the occulter and telescope satellites are aligned exactly in the direction to the object of interest. Motion that maximizes the resolution of a satellite interferometer is much more complex. This study concentrates on determining the optimal satellite motion within interferometric arrays in multi-body regimes. However, the general construct could be applied to constellations with varying imaging architectures. The resolution of an image produced by an interferometric spacecraft array is largely determined by the corresponding coverage of the (u, v plane. In turn, coverage of the (u, v plane is derived based on several characteristics of the spacecraft configuration and motion in physical space. Therefore, to determine formation motion that may be most advantageous to imaging, a mathematical model relating spacecraft motion in physical space to coverage of the (u, v plane, and thus image reconstruction, is necessary. This study employs an imaging model presented, in detail, by Millard and Howell. 9 In 98 and 99, respectively, P.H. van Cittert and F. Zernike proved that the irradiance pattern (image of an extended incoherent source is related to the Fourier transform of a function called the complex mutual coherence function. This result is the basis for interferometric imaging. Consider the situation depicted in Figure. A celestial body, S, is emitting incoherent light from some point in space. This source is approximately located in a two-dimensional plane, at some long distance, z s, from the observation point. This source plane is denoted with an (x s, y s coordinate frame. A spacecraft interferometer is located in the observation plane, labeled with an (ξ, η coordinate frame. Two apertures, that collect light photons, are located within the observation plane. These two apertures are located at points (ξ, η and (ξ, η. The goal is to construct the irradiance pattern (image, (x s, y s, of S with measurements of the light source in the (ξ, η plane. S x s y s z s (ξ, η η (ξ, η Figure : Interferometric imaging architecture. ξ

3 From the van Cittert-Zernike result, I( x s,y s µ ( u,ve iπ ( ux s +vy s dvdu [] u v where I(x s, y s is the image of the source S and the quantities u and v are defined as u ξ ξ z s Λ v η η z s Λ [-] Then, (u,v is the complex mutual coherence function measured by the interferometer and Λ is defined as the wavelength of emitted light. From equation [], it is apparent that the image is constructed from µ and therefore, depends on both u and v. In turn, u and v are functions of the relative locations of the spacecraft apertures in physical space. The (u, v plane, however, is not a physical plane. Rather, it is constructed from the relative distances of one aperture with respect to the other. In reality, the interferometer will not be located in a plane perpendicular to the line of sight to the object of interest. Thus, motion of the apertures is generally projected onto this perpendicular plane. To simulate interferometric image reconstruction, a digitized monochromatic picture is selected to represent the actual image, or a truth model. In practice, the actual image is the irradiance pattern if an infinite number of measurements in the (u, v plane are possible. The simulated image is constructed as a matrix of numbers between zero and one, with each number representing a different shade of the image color. For example, Figure is comprised of x pixels. It is represented by a x matrix with entries varying between zero and one, depending on the shade of the individual pixel. Next, a twodimensional Fast Fourier Transform (FFT of this true image matrix is computed numerically. The FFT provides the Fourier amplitude associated with each combination of spatial frequencies in the image. In other words, the D FFT is the mutual coherence function of the actual image. Figure : Digitized monochromatic picture is the true image of the observed planet Optimal distributions of measurements on the (u, v plane may be mission dependent. For example, a random distribution of measurements may capture a large variety of amplitudes. However, a aussian distribution of measurements, with a higher concentration of measurements at the center of the (u, v plane, ensures the coarse features of an image are captured while still obtaining some level of detail, as is apparent in Figure. In the following analysis, a aussian distribution is assumed to be the desired distribution. However, this choice is arbitrary and the analysis could be performed with different desired distributions. The probability density function of a aussian distribution in two-dimensions can be expressed in the form F( u( ξ,η,t,v ( ξ,η,t = e πσ u σ v u ( ρ σ + v u σ ρuv v σ u σ v [4] where σ and σ are the standard deviations of the u v distribution along the u and v directions, respectively, and ρ is the correlation between u and v values. These parameters can be changed to affect distribution characteristics on the (u, v plane. For example, as the correlation is decreased, the distribution of points on the (u, v plane becomes more random; as the standard deviations are changed, the distribution becomes more or less concentrated at the center of the plane. Therefore desired values for each parameter may also be mission dependent. DYNAMIC MODEL: CRBP Because many of NASA s imaging formation design concepts place the array near one of the collinear libration points, motion of satellites within the formation is modeled via the CRBP. The motion of a particle (spacecraft in the circular restricted threebody problem (CRBP is described in terms of rotating coordinates relative to the barycenter of the system primaries. For this analysis, the emphasis is placed on formations near the libration points in the Sun-Earth/Moon system, since these tend to be particularly advantageous for interferometric imaging. In this frame, the rotating x-axis is directed from the Sun towards the Earth-Moon barycenter. The z-axis is normal to the plane of motion of the primaries, and the y-axis completes the right-handed triad. Let, Χ = then the general non-dimensional equations of motion, relative to the system barycenter, are of the form && x y& x = [ x y z x& y& z& ] ( µ ( x + µ µ ( x ( µ r r [5]

4 measurements measurements 6 measurements 9 measurements x x -5 Figure. The reconstructed image approaches the actual image as more points in the (u, v plane are measured. These points are selected to represent a aussian distribution. where ( µ y y y x y µ & + & = [6] r r && z = ( µ z µ z r r ( x + + y / z [7] r = µ + [8] ( x ( + y / r = µ + z [9] iven a solution to the nonlinear differential equations, linear variational equations of motion in the CRBP can be derived in matrix form as δ Χ & [] ( t = A( t δχ( t ( [ ] T where δχ t δx δy δz δx& δy& δz& represents variations with respect to a reference trajectory. enerally, the reference solutions of interest are not constant. In this analysis, the reference trajectory is periodic. In particular, the focus of this work is the family of periodic halo orbits. Thus, A(t is periodic and time-varying, of the form The quantity µ is a non-dimensional mass parameter associated with the system. For the Sun-Earth system, µ A more compact notation incorporates a pseudo-potential function, Ω, such that Ω = ( µ µ + + ( x + y [] r r A( t = Ω Ω Ω xx yx zx Ω Ω Ω xy yy zy Ω Ω Ω xz yz zz [] Then, the scalar equations of motion are rewritten in the form = y& + Ω = f ( x, x&, y, y&, z z& [] & x, x x ( x, x&, y, y&, z z& & y = x& + Ω = f, [] y y ( x, x&, y, y&, z z& & z = Ω = f, [] z z where the symbol Ω j denotes Ω / x. j Equations [5]-[7] or equations []-[] comprise the dynamical model in the circular restricted three-body problem. The partials Ω are also functions of time, although kl the functional dependence on time, t, has been dropped in the notation. The general form of the solution to the system in equation [] is ( t = Φ ( t, t δχ ( t δ Χ [] where Φ(t,t is the state transition matrix (STM. The STM is a linear map from the initial state at the initial time t to a state at some later time t, and, thus, offers an approximation of the impact of the variations in the initial state on the evolution of the trajectory. The STM must satisfy the matrix differential equation 4

5 ( t, t = A( t Φ( t t Φ& [4], given the initial condition (, t = Ι 6 Φ t [5] where I 6 is the 6 6 identity matrix. It is clear that the STM at time t can be numerically determined by integrating equation [4] from the initial value. Since the STM is a 6 6 matrix, propagation of equations [4] and [5] requires the integration of 6 first-order, scalar, differential equations. Since the elements of A(t depend on the reference trajectory, equation [] must be integrated simultaneously with the nonlinear equations of motion to generate reference states. This necessitates the integration of an additional 6 first-order equations, for a total of 4 differential equations. DECENTRALIZED CONTROL LAW The analytical formulation of the traditional nonlinear control problem does apply to infinitely larger satellite formations. However, due to limitations in computational capability, a solution cannot be found for formations of satellites containing more than two spacecraft in a reasonable amount of time. As more spacecraft are added to a formation, the formulation of the traditional nonlinear optimal control problem becomes increasingly more complex. The high complexity inherent in the analysis motivates a change in either the analytical formulation of the problem or a transition to more robust computational algorithms and utilities, such as FORTRAN, UNIX, and parallel processing. This section explores the possibility of decentralizing the optimization process to reduce complexity in the analysis. Employing decentralized control and agent-based modeling techniques to solve for the optimal motion of spacecraft formations replaces one large computationally intractable problem with many smaller tractable problems. Instead of solving for the optimal control effort as a function over all time, the motion of each spacecraft is determined over discrete intervals. At every discrete interval, an optimal impulsive maneuver is defined for each spacecraft based on two competing objectives: ( use a minimal amount of control effort and ( create the best possible image at the end of the maneuver. Different constraints may be placed on spacecraft moving within the formation, including a definition of maximum available thrust, minimum available thrust, or maximum spacecraft separation from a nominal path. In this analysis, it is assumed that the discrete time interval, t disc, is fixed. It is also assumed that spacecraft are near a nominal path, defined as a halo orbit in the circular restricted threebody problem defined in equations [5]-[7]. Definition of Constraints The agent-based algorithm is initiated by employing any hard constraints on the spacecraft motion or control to define the set of locations to which each satellite can feasibly move within the defined discrete interval, t man. For example, if a satellite is constrained by a maximum available impulsive thrust of T max, and completes a maneuver in a fixed time, t man, the feasible set of points includes all such that, ( x, x&, y, y&, z z u ( t [6] & x = f,& x + ( x x&, y, y&, z z u ( t x & y = f,, & [7] y + ( x x&, y, y&, z z u ( t & z = f,,& [8] z + where T max u x ( t T max, T max u y ( t T max, and T max u z ( t T max. Because the control is impulsive, u x ( t, u y ( t and u z ( t may each be modeled as a Dirac delta function. iven initial conditions that are positioned along a halo orbit in the CRBP, the set of feasible points lies within the volume in Figure 4. The color of each point in the plot is a measure of the V required to reach that point: red indicating a high V and blue indicating a low V. The black line z y x z x x 6 y x.48 x Figure 4: Feasible points at the end of the maneuver time (t man determined by a constraint on V. 5

6 represents a nominal halo orbit near L. Similarly, if a spacecraft must remain within some maximum distance, b max, of a nominal orbit, the feasible set of positions (at the end of t man is simply all points that satisfy, x( t man x n ( t man y( t man y n ( t man z( t man z n ( t man b max [9] where x(t, y(t, and z(t are solutions to the differential equations of motion in [6], [7], and [8]. iven this position constraint, that is, b max = 5 km, and initial conditions along a nominal halo orbit, the set of feasible points lie within the sphere in Figure 5. The sphere appears superimposed onto the feasible set of points described in Figure 4. This further illustrates that different objectives may be combined to produce increasingly complex feasible sets. If a spacecraft is constrained by a large minimum available thrust, T min, and is required to remain within a specific distance from a nominal path, the feasible set may be defined as points that satisfy both equation [9] as well as equations [6], [7], and [8] where ( t u y ( t u z ( t [ u x ] T. These two constraints min create a hollow feasible region in -dimensional space. The constraint on the thrust determines the radius of the inner edge of the volume, while the constraint on position determines the radius of the outer edge. Thus, depending on the values of T min and b max, this set may be empty. One advantage of an agent-based approach is that it is relatively easy to incorporate many different constraints into the problem formulation. For example, collision avoidance may be included in the algorithm by adding a constraint that no satellite may be within a specific distance of another satellite, i.e., ( x j ( t man ( y j ( t man ( z j ( t man x i t man y i t man z i t man D [] for all i j, where i and j indicate different satellites in the array and D is some minimum separation distance. Once the feasible set is defined, evenly distributed points are identified across the set. For this study, the constraints indicate a feasible set in three-dimensional space. Thus, defining evenly dispersed points over the set is relatively straightforward. If more complex objectives are defined, resulting in feasible sets in higher dimensions, a more complex approximation would be required. In this analysis, 5 points are selected to approximate each set. In some cases, it may be advantageous to formulate a constraint as an objective function. In this case, instead of enforcing a hard constraint by limiting the feasible set, a weighting function is defined that is large near the constraint boundary and becomes smaller as points are further away from the boundary. This option is explored further in the next section. Competing Objectives After a feasible set of points is determined, each point in the set is evaluated based on the defined objectives. Objective (: Minimize Fuel The amount of fuel required to reach each of the 5 points in the feasible set may be approximated via a straightforward targeting scheme. A targeting method is presented in detail in Howell [7] but is only briefly described here. ( iven the reference halo orbit, Χ n t and a spacecraft with initial state and final state, a variation in the initial velocity is sought to shift the final state to a desired point in the feasible set. The shift is accomplished in an iterative differential corrections process. At each step, the linearized variational equations [] relative to the nominal path are used to estimate an appropriate variation, δχ ( t. The result is propagated to a final state and the process repeats until the target is reached to within a specified tolerance. The linearized equations relating initial to final variations are, δχ ( t + t Χ ( t + tmsn Χ( t Χ ( t ( t + Χ & ( Χ( t + t t + t ( t man man man man = Χ, n δ [] Figure 5: Set of feasible points at the end of the maneuver time (t man determined by a constraint on position. where ( Χ ( t Χ n t Χ t + t man ( = Φ( t + t man,t [] 6

7 and the state transition matrix is evaluated along the reference orbit. The desired change in the final state may be expressed as: δχ ( t + t min = Χ T ( t + t man Χ ( t + t man. With this substitution, an estimate for the initial variation in velocity can be computed. Using this targeting scheme, the change in velocity is determined to reach any of the 5 points in the feasible set. Figure 4 includes an example of the velocity change requirements to reach the feasible points. The color associated with each point indicates the required velocity change with red being the highest velocity change and blue being the lowest. Thus, if only one objective is important, each spacecraft would move to the point that requires the least amount of fuel expenditure (a blue point along the natural solution. However, the second objective of high-resolution imaging also affects the determination of the optimal formation configuration. Objective (: Maximize Resolution of an Interferometric Array As demonstrated in Figure, the desired distribution of points over the (u, v plane for high-resolution imaging is often aussian. The locations of points on the (u, v plane are determined by the separation of satellites in physical space (among other factors, as apparent in equations [] and []. Thus, to maximize resolution, an individual satellite should move to a point in the feasible set that creates a distribution on the (u, v plane (resulting from the relative motion of all satellites in the formation as close to aussian as possible. iven a distribution of points on a plane, various central moments may be defined to characterize the shape of the distribution. For example, the first moment of the distribution is usually denoted as the mean, the second moment as the variance, the third moment as the skew, etc. For a aussian (normal distribution, each of the moments has a specific value and definition, as summarized in Table. Ideally, as more and more points populate the (u, v plane, the values for the central moments of the distribution should approach those of a uassian normal distribution. Points on the (u, v plane are symmetric. Thus, the odd central moments will already have the required value: zero. The configuration of satellites that minimizes the difference between the values of the even moments for a aussian normal distribution and those moments of the (u, v distribution (at the end of the maneuver time is considered optimal for that maneuver. In other words, the ideal configuration minimizes the cost function, f g = + M + M u v norm( I C ( t + t + K ( t + t + K ( t + t + x dist man man man u v u ( t + t 5 + M ( t + t 5 + M ( t + t 5 + L 6 man 6 man 8 man v ( t + t 5 8 man [] where C dist (t +t man is the covariance matrix of the (u, v distribution at the end of the maneuver, K u (t +t man is the kurtosis of the points in the u-direction, K v (t +t man is the kurtosis of the points over the v- direction, and M n dist (t +t man is the nth moment in the dist-direction. An infinite number of higher moments can be included in the cost function; however, it was observed in this analysis that including moments higher than M 8 did not noticeably change the results of the simulation. In Figure 6, the evolution of a distribution is evident by adding points that minimize f g. Initially, points are chosen (randomly, but symmetrically on the plane. As more points are added to the distribution ( then 5, the characteristics of the distribution approach those of a aussian distribution. L Table : Moments of a aussian distribution Number Raw Moment Central Moment µ (mean µ +σ σ (variance µ +µσ (skew 4 µ 4 +µ σ +σ 4 σ 4 (kurtosis 5 µ 5 +µ σ +5µσ 4 6 µ 6 +5µ 4 σ +45µ σ 4 +5σ 6 5σ 6 7 µ 7 +µ 5 σ +5µ σ 4 +5µσ 6 8 µ 8 +8µ 6 σ +µ 4 σ 4 +4µ σ 6 +5σ 8 5σ 8 7

8 Figure 6: A aussian distribution is populated by identifying points on a plane that minimize f z To evaluate f g, each baseline (or set of two spacecraft in the formation must be considered; the function f g must be evaluated for every possible configuration of satellites within their feasible sets. For example, if five satellites are in the formation (ten baselines and each satellite can move to 5 different points in the given maneuver time, potentially 5 5 configurations must be evaluated. However, only the relative position of the satellites in the formation impacts the location of points on the (u, v plane. Although the satellites may be in different locations in space, if the relative distance is the same, the same points are added to the (u, v distribution. This strategy significantly reduces the number of configurations that must be evaluated, since the feasible set is selected as evenly distributed points in physical space. Objective (: Including a constraint as an objective function For some applications, it may be necessary to include a constraint in the cost function (as an objective, rather than defining a feasible set based on that constraint. Incorporating a constraint is accomplished by choosing a function (S agent with a high value near the constraint boundary and relatively low value, or zero value, away from the boundary. Traditionally an exponential function is employed such that the value of the function is equal to infinity exactly on the defined constraint boundary. Algorithm: Agent-Based Model Ultimately, a satellite (or agent moves to the point in the feasible space that minimizes a weighted sum of the objectives, subject to any constraints on the motion or control of the satellites. This weighted sum may be represented as, ( ( ( min J = c f u v + c V u, v c S u v [4] agent g, m + agent, where c, c, and c are weights selected such that each term in J agent is of the same magnitude. Of course, depending on the desired outcome of the simulation, the weights may be adjusted to encourage either less fuel usage or higher resolution. An example of the functions f g, V, and S agent is represented in Figure 7. The algorithm proceeds as follows for each satellite: ( A feasible set is defined that describes the total area the satellite can reach by the completion of the discrete maneuver time. This set is limited by any constraints on the satellite fuel usage or motion. The feasible space is sampled to form a representative subset. ( A targeting scheme is used to determine the impulsive V to reach each point in the feasible subset. ( At each point in the feasible subset, all possible relative configurations are explored and a value of f g is determined for each configuration. (4 Satellites move to the new points in their respective feasible subsets that minimize J agent. (5 The process is repeated until the end of the observation period. J agent = c + c + c Figure 7: raphical representation of individual satellite cost function for each discrete interval. 8

9 Y I Figure 9: Corresponding (u, v plane coverage for satellite motion in Figure 8. v [/km] V [km/s ] X I Figure 8: Motion of two-satellite formation with a defined maximum thrust value but no position constraint x u [/km] x -7 Figure 9: Corresponding (u, v plane coverage for satellite motion in Figure 8..5 x Time [hours] RESULTS Three different constraint combinations are simulated with the above algorithm, including scenarios with: ( a hard upper bound on the available V, ( a defined maximum satellite separation with collision avoidance, and ( a hard upper bound on the available V with a position constraint included as a competing objective. This final constraint combination may be compared with results from a traditional nonlinear control formulation, described in the next section and originally published in []. Each scenario may be evaluated for any number of satellites. However, for comparison purposes, scenario ( is simulated with only two satellites in the formation. Also for comparison purposes, scenario ( only includes the planar dynamics in the CRBP. Scenarios ( and (: Several assumptions are incorporated in the agentbased simulations. It is assumed that the first measurement occurs when the satellites are initially positioned within the formation and a maneuver is implemented every minutes. It is also assumed that the object of interest is always located in a direction perpendicular to the inerial X I -Y I plane. In scenario (, the maximum V is 5 mm/s in each of the x, y, and z directions. In scenario (, satellites must also remain within 5 kilometers of the nominal orbit. The position of satellites in scenario ( over a 5.5- hour observation period appears in Figure 8, with blue indicating satellite and red indicating satellite. Each star reflects the position of a satellite at the end of the specific discrete time interval. The position is plotted relative to the nominal halo orbit, projected onto the intertial X I Y I plane. Initially, the two satellites in the formation are placed near the nominal halo orbit, separated by 5 meters. As time progresses, the satellites move away from the nominal orbit (green circle near the center of the plot; there is no constraint on the position of the satellites in this scenario. Thus, in the final configuration, the satellites are approximately 5 km apart. Figure 9 displays the corresponding (u, v coverage. The distribution is close to aussian with several points clustered near the center of the plane and fewer points near the edges. The weights in this scenario are selected such that c = c = c =, which implies the algorithm favors high resolution imaging over fuel economy. Finally, the V required for each satellite in the formation over the duration of the mission is summarized in Figure. Because a maximum available thrust is included as a constraint, no maneuvers require more than 5 mm/s. In Figure, blue and red indicate the V used by satellites and, respectively. Figure : Impulsive velocity change coverage for satellite motion in Figure 8. 9

10 Y I [m] X I [m] v [/km] Figure : Motion of two-satellite formation with a defined maximum thrust value and position constraint. V [km/s ] 8 x u [/km] x -4 Figure : Corresponding (u, v plane coverage for satellite motion in Figure. x Time [hrs] Figure : Impulsive velocity change coverage for satellite motion in Figure. The position of the satellites is plotted Figure displays the motion of four spacecraft in scenario ( with blue, red, green, and pink representing satellites,,, and 4, respectively. In this scenario, the satellites must remain separated by at least 5 meters, with a maximum satellite separation of kilometers. The points are relative to the nominal orbit, again projected onto the inertial X I -Y I plane. Satellites in the formation are initially placed along the nominal orbit in a string-of-pearls configuration, separated by 5 meters. The satellites move away from the nominal orbit, but remain within 5 meters of the nominal orbit for the entire 5.5 hour observation period. The resulting aussian distribution on the (u, v plane appears in Figure and Figure represents a summary of the required impulsive V maneuvers. There is no constraint on the thrust level in scenario (. Thus, thrust levels are extremely small; these levels are not feasible via current thruster technology. Scenario (: Comparison to Traditional Nonlinear Control Problem The results from scenario ( are compared to the results with a traditional nonlinear controller. The nonlinear control problem is formulated to minimize fuel usage while maximizing image resolution. This analysis was first presented by Millard and Howell [] and is summarized for comparison in this study The goal of this continuous nonlinear minimization process is to create a distribution on the (u, v plane that is as close to aussian as possible, given a specific total measurement time ( t f = t, man formation size, and with minimum cost in terms of thrust. Therefore, the cost function may be represented as J t = f U b num ( t w F( u ( x y, z, v ( x, y, z i= i, dt [5] where w is a weighting parameter and U ( t is the magnitude of the total control effort for all spacecraft in the formation, i.e. U [ ] ( t = U ( t U ( t U ( t L U ( t U ( t U ( t x y z xj yj zj ( t = U ( t + U ( t + U ( t + L + U ( t + U ( t + U ( t U x y z i xj yj zj [6] The first subscript, d, on the control variable, U dp, describes the direction of the control being implemented (x, y, or z while the second subscript, p, indicates the particular spacecraft (,,,, or n for nominal on which the control is implemented. Again, F is a function representing a aussian distribution over the (u, v plane, as defined in equation [4]. The trajectory optimization process occurs over a fixed observation time, t f.

11 The optimization process continues, subject to constraints on the spacecraft motion. One type of constraint that may be employed is a maximum baseline length, i.e. S S S ( t = ( x x + ( y y + ( z z n n n ( t = ( x x + ( y y + ( z z j max ( t = ( x x + ( y y + ( z z b j M n n j n n j n n b max b max [7] The constraints, S j t, ensure that the maximum baseline, or maximum distance between any two spacecraft, remains below b max and that the formation remains bounded relative to the specified libration point orbit. The values of the variables x, y, and z, shift subject to the dynamics in the planar CRBP, as defined by, && x y& x = && y + x& y = ( ( µ ( x + µ / ( x + µ + y ( ( µ ( x + µ + y µ ( x ( µ ( x ( µ y / + [8] µ y / ( y ( x ( µ + y / [9] The optimization problem is solved numerically via a sequential quadratic programming method for two spacecraft moving near a nominal halo orbit. Initial conditions for the algorithm are selected based on knowledge of the phase space surrounding the nominal orbit. Details of this nonlinear optimal control problem derivation as well as the numerical solution is available in Millard and Howell. This small formation is used to construct an image of a Jupiter-like planet located 4. light years from Earth (a planet near our closest star, Alpha Centauri at a wavelength of 5 nm (visible blue light. The spacecraft trajectory, corresponding coverage of the (u, v plane and simulated image reconstruction appear in Figure 4. In this example, the planet is placed in the inertial Z I -direction; therefore, the motion occurs in a plane perpendicular to the direction of the object of interest, i.e., the inertial X I -Y I plane. The location of the object of interest is arbitrary. The bottom line in Figure 4 illustrates motion of the formation in the inertial X I -Y I direction at four times during the 8 day period. Each square or triangle represents the location of a specific spacecraft in physical space corresponding to four different times at which a measurement of the mutual coherence function is obtained. The two spacecraft in the formation are initially placed meters apart on opposite sides of an L reference Lyapunov orbit. This position appears in an inertial frame, relative to the instantaneous location along the reference Lyapunov orbit. The initial velocities of the spacecraft are determined such that the state is in the center subspace of the associated L orbit. Figure 4 also illustrates the motion of the short baseline formation over the course of one halo orbit period, or approximately 8 days. The spacecraft originate near the center of the plot and move outward toward the constraint boundary. Because this problem is formulated in two dimensions, Figure 4 encompasses the entire path of the spacecraft over one libration point orbit. Along the middle line in Figure 4 is the corresponding (u, v plane coverage at each of the four times. Each circle represents a point on the (u, v plane where the mutual coherence function is measured. The coverage of the (u, v plane appears quasi-aussian. There are far more points near the center of the plane and fewer points are positioned near the edges of the constraint boundary. Finally, the top line in Figure 4 represents the image reconstruction as time progresses. This image of the Jupiter-like planet could be improved by increasing the maximum allowable baseline and/or increasing the number of satellites in the formation. A number of assumptions are incorporated into the image reconstruction methodology. First, the image is modeled as fixed, i.e., it remains unchanged throughout the measurement period of 8 days. It is also assumed that no starlight nulling is necessary. The spacecraft formation processes one measurement every.8 days. The object of interest is assumed to be stationary and located in the inertial z-direction from the formation. Finally, Figure 5 illustrates the position, (u, v plane coverage, and corresponding image reconstruction for the same scenario resulting from the agent-based model. Again, the position is plotted relative to the nominal orbit, projected onto a plane perpendicular to the line-of-sight vector to the object of interest. Initially, the two satellites in the formation are located along the X I -axis, offset from the nominal halo orbit by meters. The initial velocities of the spacecraft are determined such that the state is in the center subspace of the associated L orbit. The bottom line in Figure 5 illustrates the motion of the formation over the course of one halo orbit period, or approximately 8 days. In this scenario, the time between each maneuver, t man, is increased to.8 days for comparison purposes. As time progresses, the satellites move away from the nominal orbit and approach the constraint boundary. A maximum allowable baseline constraint (b max = kilometers is included as an objective, S agent. The center row of plots in Figure 5 represents the corresponding distribution on the (u, v plane at the end of each maneuver. Finally the top row is the resulting image reconstruction of a fictitious planet, represented by a picture of Jupiter. The general results from the two analytical methods

12 are similar, but not exactly the same. There are a few possible explanations for the discrepancies. The agent-based model only approximates the results of the complete nonlinear control problem. The approximation occurs on three levels. First, the agentbased algorithm samples the feasible space of motion with a finite number of points. Second, only the first 8 moments of a aussian distribution are utilized to determine the best imaging formations. Finally, the trajectories of the satellites within the formation are essentially sampled at discrete intervals of time, rather than simulated over a continuous time interval. Also, the choice of weights in both the nonlinear control problem (w and the agent-based model (c, c, and c may affect the results of the simulation. There is no direct relationship between the different weights in the problem formulations. Although the results are not exactly the same, agentbased analysis does posses some distinct advantages. The agent-based model offers a tool to analyze problems that may be too complex to be effectively solved by other methods. The model is validated to significant accuracy for a relatively simple example and could be an important tool for initial analysis in problems of this type. CONCLUSIONS This paper explores the use of decentralized control and agent-based modeling techniques to solve for the optimal motion of spacecraft formations with highly complex objectives, defined in multi-body regimes. The outcome of the agent-based model is compared to a more traditional non-linear optimal control solution for a relatively simple two-satellite array example. Then, taking advantage of the reduced computational requirement, the agent-based model is employed to simulate more complex arrays with increasing number of satellites and constraints. Utilizing an agent-based model may have several advantages in complex optimization problems. Because agent motion is based on simple rules, it is relatively easy to incorporate several constraints and objectives without significantly increasing computational requirements. Also, when compared to a traditional sequential quadratic programming method, the amount of numerical integration is greatly reduced. Solving problems that are intractable due to extremely high computational requirements may be feasible. In this particular problem, the agent-based model produces results comparable to those from a traditional nonlinear formulation for a two-spacecraft formation. Future work includes a more rigorous comparison of the agent-based formulation directly to the nonlinear control formulation. Also, rather than sampling a feasible set of future satellite positions, search algorithms may be developed to increase the accuracy of the agent-based simulation. ACKNOWLEDEMENTS This work was performed at Purdue University with support from NASA oddard Space Flight Center under contract numbers NCC5-77 and NN4P69. REFERENCES. TPF Science Working roup, Terrestrial Planet Finder: A NASA Origins Program to Search for Habitable Planets, endreau, K. C., W. C. Cash, A. F. Shipley, and N. White, The MAXIM Pathfinder X-ray Interferometry Mission, Proceedings of the International Society of Optical Engineering (SPIE Conference, Waikoloa, Hawaii, August 6 8,.. MacDowall, R., M. Kaiser, and N. opalswamy, Solar Imaging Radio Array (SIRA: Radio Aperture Synthesis from Space, Proceedings of the 4 th AAS Solar Physics Division Meeting, May, Vol., pp Millard, Lindsay D. and Kathleen C. Howell, Optimal Reonfiguration Maneuvers for Spaecraft Imaging Arrays in Multi-body Regimes, Acta Astronautica, vol.. 6, 8, pp Hussein, I. I., Scheeres, D. J., and Hyland, D. C., Formation Path Planning for Optimal Fuel and Image Quality for a Class of Interferometric Imaging Missions, AAS/AIAA Space Flight Mechanics Conference, February. 6. Hussein, I. I., Scheeres, D. J., and Hyland, D. C., Control of Satellite Formations for Imaging Applications, Proceedings of the American Control Conference,, pp Hussein, I. I., Scheeres, D. J., and Hyland, D. C., Optimal Formation Control for Imaging and Fuel Usage, 5 th AAS/AIAA Space Flight Mecahnics Conference, Copper Mountain, Colorado, January - 7, 5. AAS Hussein, I. I., D. J. Scheeres, and D. C. Hyland, Optimal Formation Control for Imaging and Fule Usage with Terminal Imaging Contraints, Proceedings of the IEEE Conference on Control Applications, Hussein, I. I. and A. M. Block, Dynamic Coverage Optimal Control for Interferometric Imaging Spacecraft Formations, Proceedings of the 4 rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, December 4, pp Hussein, I. I. and A. M. Bloch, Dynamic Coverage Optimal Control for Interferometric Imaging Spacecraft Formations (Part II: The Nonlinear Case, Proceedings of the American Control Conference, 5, pp Millard, Lindsay D. and Kathleen C. Howell, Control of Interferometric Spacecraft Arrays for (u, v Plane Coverage in Multi-Body Regimes, The Journal of the Astronautical Sciences, Vol. 56, No., January-March 8.. Fax, J. A, Optimal and Cooperative Control of Vehicle Formations, PhD Dissertation, California Institute of Technology,.

13 . Inhalan,., D. M. Stipanovic, and C. J. Tomlin, Decentralized Optimization, with Application to Multiple Aircraft Coordination,n the Proceedings of the 4 st IEEE Conference on Decision and Control, December.. Tabuada, P.,. J. Papps, and P. Lima, Decentralizing Formations of Multi-Agent Systems Proceedings of the th Mediterranean Conference on Control and Automation, Lisbon, Portugal, July. 4. Raffard, Robin L., Claire J. Tomlin, and Stephen P. Boyd, Distributed Optimization for Cooperative Agents: Application to Formation Flight, 4 rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, December 4-7, Campbell, Mark and Thomas Schetter, Comparison of Multiple Agent-Based Organizations for Satellite Constellations, Journal of Spacecraft and Rockets, Vol. 9, No., March-April. 6. Mueller, J. B., D. M. Surka, and B. Udrea, Agent- Based Control of Multiple Satellite Formation Flying, 6 th International Symposium on Artificial Intelligence, Robotics and Automation in Space, June 8-,, Montreal, Canada. 7. Howell, K. C., Three-dimensional, Periodic, Halo Orbits, Celestial Mechanics, Vol., 984, pp.5-7.

14 v [/km] v [/km] v [/km] v [/km] u [/km] u [/km] u [/km] u [/km] Y I Y I Y I Y I X I X I X I X I Figure 4: Simulation of two-spacecraft imaging array resulting from traditional nonlinear optimal control problem formulation. 4

15 v [/km] v [/km] v [/km] v [/km] u [/km] u [/km] u [/km] u [/km] Y I X I Y I X I Y I X I Y I X I Figure 5: Simulation of two-spacecraft imaging array resulting from agent-based mode 5

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